The generalized finite element method for Helmholtz equation

www.elsevier.com/locate/cma

ThegeneralizedfiniteelementmethodforHelmholtzequation:

Theory,computation,andopenproblems

TheofanisStrouboulis

aba,*ˇkab,RealinoHidajat,IvoBabus

aDepartmentofAerospaceEngineering,TexasA&MUniversity,CollegeStation,TX77843,USA

InstituteforComputationalEngineeringandSciences,UniversityofTexasatAustin,Austin,TX78712,USA

Received23May2005;receivedinrevisedform23September2005;accepted23September2005

Abstract

InthispaperweaddressthegeneralizedfiniteelementmethodfortheHelmholtzequation.WeobtainourmethodbyemployingthefiniteelementmethodonCartesianmeshes,whichmayoverlaptheboundariesoftheproblemdomain,andbyenrichingtheapproxi-mationbyplanewavespastedintothefiniteelementbasisateachmeshvertexbythepartitionofunitymethod.Hereweaddresstheq-convergenceofthemethod,whereqisthenumberofplanewavesaddedateachvertex,fortheclassofsmooth(analytic)solutionsforwhichwegetbetterthanexponentialconvergenceforsufficientlysmallhdependingonp.Animportantobservationisthatwecanmonitortheaccuracyinanycomputedsolutionquantityofinterestatnegligiblecostbyusingq-extrapolation.Ourresultsassumeexactintegrationofalltheemployedintegrals.Furtherstudiesareneededtoanalyzetheeffectsofthenumericalintegrations,andalsotheeffectoftheroundofferrors.

Ó2005ElsevierB.V.Allrightsreserved.

Keywords:Generalizedfiniteelementmethod(GFEM);Helmholtzequation;Exponentialconvergence;aposteriorierrorestimation

1.Introduction

ˇkaetal.[1],ThegeneralizedfiniteelementmethodoriginatedfromtheworkonthepartitionofunitymethodofBabus

ˇkaandMelenk(seeMelenk’sM.Sc.andPh.D.theses[2,3]andthepapersstemmingfromthesetheses[4–6]).LetandBabus

usalsonotethatOdenandDuartewerethefirsttoemploythepartitionofunitymethodinameshlesssettinginthecontextofthehp-cloudmethod[7,8].

Thegeneralizedfiniteelementmethod(generalizedFEMorGFEM)addressedhereisadirectextensionoftheclassicalfiniteelementmethodenrichedbythepartitionofunitymethodandwasproposedintheworksofStrouboulisetal.[9–15]forsolvingcoerciveellipticproblems(e.g.theLaplaceequation,theequationofheatconduction,etc.)inproblemswithcomplexdomainsusingenrichmentbyhandbookfunctions.Letusalsomentiontheworkontheextendedfiniteelementmethod(XFEM)byBelytschkoandco-workers[16,17]whichhassimilaringredients,andalsotheworkbyDuarteandco-workers[18].Themathematicalaspectsofthegeneralizedfiniteelementmethods,includingthemethodproposedhere,

ˇkaetal.[19–21]wheremanymorereferencescanbefound.ThereaderisalsoreferredwereaddressedintheworkofBabus

totheworksofLiszkaandOrkisz[22]andTworzydlo[23]ontheFDMwhichisafinitedifferenceanalogofthemeshlessapproach;seealsotherecentworksbyLiszkaandco-workers[24,25].

*Correspondingauthor.

E-mailaddress:strouboulis@aeromail.tamu.edu(T.Strouboulis).

0045-7825/$-seefrontmatterÓ2005ElsevierB.V.Allrightsreserved.doi:10.1016/j.cma.2005.09.019

4712T.Strouboulisetal./Comput.MethodsAppl.Mech.Engrg.195(2006)4711–4731

Thisworkfocusesonthecomputationalaspectsofthegeneralizedp-versionofthegeneralizedFEMfortheHelmholtzequationanditstheoreticalbasisisgiveninChapter8ofthePh.D.thesisofMelenk[3],wherethetheoreticalaspectsofthepartitionofunitymethodfortheHelmholtzareaddressed.OtherimportantworksalongthesamelineswerecontributedbyLaghroucheandco-workers[26–28],byOrtiz[29],andbyAstleyandGamallo[30].Thethree-dimensionalversionofthemethodwasaddressedbyPerrey-Debainetal.in[31].LetusalsomentiontherelatedworksbyFarhatetal.[32,33]onthediscontinuousenrichmentmethod(DEM)whichalsoemploysenrichmentbyplanewavefunctionsusingaformulation

`zeetal.[34–36]onthevar-basedonthediscontinuousGalerkinwithLagrangemultipliers,andalsotheworkofLadeve

iationaltheoryofcomplexrays(VTCR)whichalsousesenrichmentsoftheapproximationbylocalsolutionsofthewaveequation.

Amainfeatureoftheapplicationofthemethodsenrichedbyplanewavesolutions(thePUMofMelenk[3],Bettessand

`zeetal.[34–36])isthattheycanobtainLaghrouche[26,27],theDEMofFarhatetal.[32,33],andtheVTCRofLadeve

accuratesolutions(e.g.solutionswitherror1%orbetterinrelativevalueoftheH1-seminormoftheerror)whileusingmesheswithmeshsizehofoneorseveralwave-lengthsk=2p/k.Themainlimitationofthemeshsizehinallthesemeth-odsisduetothepollutionduetothewavenumberwhichisamaincharacteristicofGalerkinsolutionoftheHelmholtz

ˇka[37,38],thebookbyIhlenburg[39],andthepaperofBabusˇkaandequation,seee.g.theworksofIhlenburgandBabus

Sauter[40].ThepollutioneffectcanbeunderstoodasashiftintheGalerkinsolutionwithrespecttotheexactsolutionduetonumericaldispersion,andithasbeenanalyzedinthecontextofhandhpfiniteelementmethodbyIhlenburgand

ˇka[37,38].Deraemaekeretal.[41]presentedageneralapproachforassessingthepollutionofvariousapproxima-Babus

ˇkaandSauter[40]addressedtheimportantquestionifthepollutionisavoid-tionsoftheHelmholtzequations,andBabus

ˇkaetal.[43,44]addressedtheablebyspecialdesignoftheapproximationmethodinoneandhigherdimensions.Babus

pollutioneffectforfiniteelementsolutionsoftheHelmholtzwithrespecttoa-posteriorierrorestimation.Aswewillseebelow,thepollutioneffectisthemainlimitingfactorofthemesh-sizeemployedbythegeneralizedFEMfortheHelmholtzequation.

Followingthisintroduction,wesummarizetheformulationoftheHelmholtzproblemwithRobinboundaryconditions,weformulatethegeneralizedFEMenrichedbyplanewavefunctionsusingCartesianmeshes,weaddresstheimplemen-tation,andthecomputationalresultsobtainedusingthegeneralizedp-versionofthemethod,andwegiveasummaryofoutstandingopenproblems.2.Preliminaries

InthissectionweoutlinethebasicresultsfortheHelmholtzproblemwithRobinboundaryconditionsinafinitedomain.FormoredetailsandproofsthereadershouldconsultChapter8inthePh.D.dissertationofMelenk[3].

Modelproblem(Helmholtzproblem):LetX&R2,beaboundeddomain,withboundaryoX=C1[C2,C1\\C2=;,asshowninFig.1.Wewillbeinterestedinthesolutionu,oftheboundary-valueproblem

ÀDuÀk2u¼finX;ou

¼g1onC1;onou

Àiku¼g2onC2.on

ð1aÞð1bÞð1cÞ

AlthoughallthefactsmentionedbelowholdalsoforX&Rn,forsimplicityandpracticalissuesrelatedwiththenumer-icalimplementationswearefocusingonthetwodimensionalcase,X&R2.

Γ2Γ1ΩFig.1.ExampleofadomainXwithinteriorboundaryC1,andouterboundaryC2.T.Strouboulisetal./Comput.MethodsAppl.Mech.Engrg.195(2006)4711–47314713

Weakformulation(weakHelmholtzproblem):TheweakformulationoftheHelmholtzproblemisgivenby:Findu2H1(X)suchthatBðu;vÞ¼LðvÞ8v2H1ðXÞ;where

Bðu;vÞ¼LðvÞ¼

Z

X

ð2aÞ

Z

X

rur󰀂vdXÀk

2IXi¼1

2

Z

X

u󰀂vdXþik

I

C2

u󰀂vds;ð2bÞð2cÞ

f󰀂vdXþ

Ci

gi󰀂vds;

whereH1(X)isthespaceoffunctionswithsquare-integrablederivativesoverX.

Let

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidef22

jjjujjj¼krukL2ðXÞþk2kukL2ðXÞ.Wehave,

ð3Þ

Theorem1(Existenceanduniqueness).Letf2HÀ1(X),gi2HÀ1/2(Ci),i=1,2.Then,thereexistsauniquesolutionoftheHelmholtzproblemwhichsatisfies

!

2X

jjjujjj6CðX;kÞkfkHÀ1ðXÞþkgikHÀ1=2ðCiÞ.ð4Þ

i¼1

HereCðX;kÞindicatesthedependenceofConthedomainXandthewave-numberk.

Proof.ItissufficienttoconsiderthecasewithC1=;.Uniqueness(resp.existence)followsifwecanshowthat:ZZIruÁr󰀂vdXÀk2u󰀂vdXÆiku󰀂vdC¼0

X

X

C2

ð5Þ

withplus(resp.minus)signinfrontoftheboundarytermimpliesthatu=0.Choosingv=uandconsideringtheimaginary

partitfollowsthatu=0onC2.Thus,u2H10ðXÞ,andwehave

ZZ

2

ruÁr󰀂vdXÀku󰀂vdX¼08v2H1ðXÞð6Þ

X

X

whichmeansthatusatisfiestheHelmholtzequationwithf=0,andhomogeneousboundaryconditionsonoX,fromwhich

itfollows(seeMelenk[3,p.117])thatuisidenticallyequaltozero.Fortheproofof(4),seeMelenk[3,pp.118–121].hRemark1.InthecasethatXisconvex,Cdoesnotdependonk.

InwhatfollowswewillassumethatXisanannulusandwewillletf=0.3.GeneralizedFEMfortheHelmholtzequation

h/Di;i

LetDhbeauniformmeshofsquaresofsizehcoveringthedomainX,asis,e.g.,showninFig.2(a).Let

¼1;...;nnodes,betheclassicalpiecewisebilinear‘‘hat’’functionsassociatedwiththenodeslocatedattheverticesofthesquares,andlet

ÈÉDhDhhdef

xD¼supp/¼x2Xj/ðxÞ>0ð7Þiii

nnodes

Dhh

bethesupportof/Diwhichconsistsofthefoursquareswhichsharethenode.Thenf/igi¼1

hnnodes

unitysubordinatetothecoverfxDsatisfyingigi¼1

nnodesXi¼i

h

/DonX;i󰀆1

h

k/DikL1ðR2Þ61;

h

kr/DikL1ðR2Þ6

isaLipschitzpartitionof

C

h

ð8Þ

Wewillconstructthespaceofgeneralizedfiniteelementsolutionsbycombiningthefollowingdiscretespacesoffunc-tionsonthemeshDh:

4714T.Strouboulisetal./Comput.MethodsAppl.Mech.Engrg.195(2006)4711–4731

(a)(b)Fig.2.(a)ExampleofaCartesianmeshDhusedintheformulationofthegeneralizedfiniteelementmethodoveranannulardomainX.(b)Atypicalhexplodedviewofthehatfunction/Dioveritssupport.(a)Thestandardbi-p(tensorproduct)finiteelementspace

󰀅ÈÉp0p󰀅~SDh¼v2CðXÞvjs󰀇Fs2Q;

ð9Þ

episthespaceofbi-p(tensorproduct)polynomialfunctionsofdegreepdefinedover^´andBa-s(seeinSzabowhereQ

2

ˇka[42,p.97]),andFs:^buss¼ðÀ1;1Þ!1,istheaffinetransformationh

^xx¼Fsð^xÞ¼xsþmid

22

mapping^s¼ðÀ1;1Þintothesquaresofsizeh,centeredatxsmid.(b)Thepartitionofunityspace

󰀅)(

󰀅nnodesX󰀅;qk;qDhh

Wkv¼/DDh¼ivi󰀅vi2WlocðxiÞ;󰀅i¼1

whereW

k;qloc

ð10Þ

ð11Þ

󰀆󰀆󰀇󰀇'

2pn2pn

þysin¼spanwk¼expikxcos;n¼0;...;qÀ1qq

&

ð12Þ

nnisthelocalspaceoflinearcombinationsofplanewavestravellinginthedirectionsðcos2p;sin2pÞ,n=0,...,qÀ1.qq

hFig.3depictstheemployedwavedirectionsforq=1,3,5,7,9,and11,inatypicalpatchxDi.

hFig.3.ExamplesoftheemployedwavedirectionsinatypicalpatchxDi.T.Strouboulisetal./Comput.MethodsAppl.Mech.Engrg.195(2006)4711–47314715

Generalizedfiniteelementsolution:Find

!qnfemnnodesXXXðiÞðiÞ;q;qh

upbkNkþ/DajWj2Skh¼h;pi

k¼1

i¼1

j¼1

ð13Þ

suchthat

;q;q

8v2SkBðuph;vÞ¼LðvÞh;p;

ð14Þ

whereNkdenotesastandardFEbi-pbasisfunctionsonDh,nfemisthetotalnumberofdegreesoffreedomforthebi-pFEðiÞh

basisfunctions,andWj;j¼1;...;qaretheemployedplanewavefunctionsinthepatchxDi.

Thestabilityandquasi-optimalityofthegeneralizedFEMisestablishedfromthefollowingresultswhichwereformu-latedandproveninMelenk[3].

Theorem2(Stabilityandquasi-optimality).LetXbeastar-shapeddomainwithsmoothboundary,andletSbediscreteapproximationspacessatisfying

󰀊󰀋󰀊󰀋

2

infkvÀvkL2ðXÞþhkrðvÀvÞkL2ðXÞ6CðXÞhjvjH2ðXÞþð1þkÞjjjvjjjð15Þ

v2S

forallv2H2(X),withCindependentofvandh.ThenthereexistC1;C2;C3>0,dependingonlyonX,suchthatundertheassumption

ð1þk2Þhwehave

RBðu;vÞC2

Pinfsup

u2Sv2Sjjjujjjjjjvjjj1þkand

;qjjjuÀuphjjj6C3infjjjuÀvjjj.

v2S

ð16Þð17Þ

ð18Þ

h

Proof.SeeMelenk[3,pp.122–123].Wealsohave:

Lemma1(ApproximationpropertyofgeneralizedFEMspace).Thegeneralizedfiniteelementspace

;qpk;qSkh;p¼SDhÈWDh

def

ð19Þ

satisfies(15).

Proof.TheprooffollowsfromMelenk[3,p.138].

h

ItfollowsthatTheorem2isapplicableinthegeneralizedFEM.

Remark2.Atpresent,wecannotsayaprioriwhatissufficientlysmallh,andwhatisthevalueofC3.

Remark3.TheconstantC3in(18)isrelatedtothepollutionduetothewavenumber,whichisdirectlyrelatedtothenumericaldispersion.ThemaineffectisthatthewavenumberoftheGalerkinsolutionisdifferentfromthewavenumber

;q

oftheexactsolution.ThepollutioninthesolutionuphoftheHelmholtzequationcanbemeasuredbydividingitserror;qjjjuÀuphjjjbythecorrespondingerrorinthebestapproximationinfv2Sk;qjjjuÀvjjj.Thedeviationofthisratiofromoneh;p

measuresthepollutionwithrespecttothejjjÆjjjnorm.Similarly,thepollutioncanbemeasuredinothernorms,e.g.thekÁkL2norm,etc.,andalsoinanyoutputFðuÞofinterest.

Remark4.Intheone-dimensionalcaseandforfiniteelementapproximationusingpolynomialshapefunctionsofdegree

ˇka[37,38]haveproventhatp,IhlenburgandBabus

󰀆󰀇p;0

krðuÀupkhhÞkL2ðXÞ

%C1þC2k;ð20Þ

pinfkrðuÀvÞkL2ðXÞ

v2Sh;p

;0kh

whereweletSh;p¼Skh;p.ThetermC2kðpÞreflectsthepollutionduetothewavenumber.Tounderstanditseffectlet

󰀆󰀇1ptolp;ð21Þh¼

kC1

p

4716T.Strouboulisetal./Comput.MethodsAppl.Mech.Engrg.195(2006)4711–4731

wheretolisadesirederrortolerance.Thenfortheerrorinthebestapproximationwehave

v2Sh;p

infkrðuÀvÞkL2ðXÞ%tol

ð22Þ

;0

foranyk,whileforthesamechoicesofhandp,fortheerrorintheGalerkinsolutiontotheHelmholtzproblemuph,wehave

󰀆󰀇

C2;0

k;ð23ÞkrðuÀuphÞkL2ðXÞ%tol1þC1

whichdivergestoinfinitylinearlywithk.

;q

Eq.(18)relatestheerrorinthegeneralizedFEM,jjjuÀuphjjj,withtheerrorinthebestapproximationinfv2Sk;qjjjuÀvjjj.h;p

Variousstatementscanbemadeabouttheconvergenceoftheerrorinbestapproximationdependingontheregularityofthesolutionu.Inthispaperwewilladdressonlythecaseofanalyticsolutionsforwhichwehave:

󰀂¼X[oX.Then,Theorem3(handpandq-convergence).LetthesolutionuoftheHelmholtzproblem(2c)beanalyticonX1.Forfixedhwehave

k;q

v2Sh;p

infjjjuÀvjjj6CðhÞeÀðaqþbpÞ;

ð24Þ

wherea,balsodependonh.2.Forfixedpandqwehave

infjjjuÀvjjj6Cðp;qÞhÀp.

v2Sh;p

k;q

ð25Þ

ThemainresultsneededfortheproofaregiveninMelenk[3,Sections8.4.–8.6.].

;q

Remark5.When(1+k2)hissufficientlysmall,Theorem3alsocharacterizesjjjuÀuphjjj,theerrorinthegeneralized;qp;qFEMsolutionuph.Whenthisassumptiondoesnothold,theerrorinuhcanberatherlarge,e.g.therelativeerror;qjjjuÀuphjjj=jjjujjjcanbe100%orlarger.

Remark6.Followingasimilarapproachasin[41,40],itisalsopossibletocharacterizethedependenceofthepollutiononpandqinasimilarformasin(20).4.Implementationaspects

Theaimofthecomputationistoobtainreliableinformationforoutputsofinterest.Forexample,wemaybeinterested

;q;q

insmallerrormeasuredintheH1-seminormkrðuÀupxÞÀupxÞj,hÞkL2ðXÞ,orinsmallerrorinthepressureatapointjuð󰀂hð󰀂p;q

thevalueofanintegralofthepressure,oringeneralFðuÞÀFðuDhÞ,whereFisanoutputfunctional.Theimplementationconsistsofthefollowingstages:1.FormingthediscreteGFEM(generalizedFEM)equations

KU¼F.

ð26Þ

Inpractice,thecoefficientsinKandFarereplacedbyapproximatevaluesevaluatedbynumericalquadrature,andhence

;q

wemustaddresstheeffectoftheerrorinthenumericalintegrations.Moreprecisely,letuph;ÃbethediscreteGFEMsolu-tionobtainedusingtheperturbedsystemK*U*=F*,whereK*,F*are,respectively,theperturbedstiffnessmatrixandloadvectorobtainedusingnumericalquadrature.Then,usingthetriangleinequalitywehave󰀅󰀅

󰀅jjjuÀup;qjjjÀjjjup;qÀup;qjjj󰀅6jjjuÀup;qjjj6jjjuÀup;qjjjþjjjup;qÀup;qjjj.ð27Þhhh;Ãh;Ãhhh;Ã

;qp;qThetermjjjuphÀuh;Ãjjjmeasurestheeffectofthenumericalintegrationerrors,andcanaffecttheconvergenceofthemethod.ForananalysisofthenumericalintegrationmethodsforaclassofmeshlessGalerkinapproximationsof

ˇkaetal.[45].ThemaintheLaplacian,seeBabusconclusioninthisworkisthat,itisimportantforthenumericalinte-PNDOF

grationtosatisfytheconsistencyconditionj¼1Kij¼0,namelythattherowsumofthestiffnessmatrixiszero.2.SolvingthediscreteGFEMequationsbyalinearequationsolver.WeemployGausseliminationwithpartialpivoting,andhencewemustalsoaddresstheeffectoftheroundofferror.

T.Strouboulisetal./Comput.MethodsAppl.Mech.Engrg.195(2006)4711–47314717

;q

3.AposteriorianalysisoftheerrorinanoutputFðuphÞ.Herewewillusearatherstraightforwardextrapolationapproachtoestimatetheerrorinanyoutputofinterest,byemploying

;qp;qþs;q

FðuÞÀFðupFðuhÞÀFðuphÞhÞ%.p;qþsFðuÞFðuhÞ

;q

UsingsufficientlyhighswecanobtainareliableestimatorforanycomputedoutputFðuphÞ.

ð28Þ

5.Computationalresultsandtheiranalysis

InwhatfollowswewillpresentandanalyzethecomputationalGFEMresultsobtainedbysolvingtheproblemofascat-teringofaplanewavebyarigidcircularcylinder,whichisdepictedinFig.4(a),usingthe(h,p,q)-versionofthegeneralizedFEM.Employingcylindricalcoordinates(r,h),theboundaryvalueproblemforthescatteredpressureu(r,h)reads

r2uþk2u¼0;r>a;ouo

¼ÀðuincÞ;r¼a;oror󰀆󰀇

pffiffiou

Àiku¼0;limr

r!1orwhere

uincðr;hÞ¼P0eikrcosh

ð30Þð29aÞð29bÞð29cÞ

isthepressurefieldfortheincidentplanewave.Usingseparationofvariables(see,e.g.,Jones[46,p.412])weobtainthescatteredfieldintheform

10X

nJnðkaÞHnðkrÞcosðnhÞ;ð31Þ󰀂niuðr;hÞ¼ÀP00HðkaÞnn¼0where󰀂0=1,󰀂n=2,n50,Hn(z)isthecylindricalHankelfunctionofthefirstkind,andJn(z)isthecylindricalBesselfunc-tionofthefirstkind.Using

g1¼À

o

ðuincÞonC1¼Cson

ð32Þ

intheNeumannboundarycondition(1b),and

g2¼

ou

Àikuon

onC2

ð33Þ

intheRobinboundarycondition(1c),wehavecompletedthedefinitionofourmodelexampleinthefiniteannulardomainXboundedbyC1andC2.

Fig.4.(a)Notationsusedinthedefinitionofthemodelexampleofscatteringofaplanewavebyarigidcircularscatterer.(b)Contoursoftherealandimaginarypartsofthescatteredfield.

4718T.Strouboulisetal./Comput.MethodsAppl.Mech.Engrg.195(2006)4711–4731

Inourcomputationsweemployedthefollowingdata:••••

Radiusofscatterer,a=1;

Amplitudeofincidentwave,P0=1;RadiusoftruncationboundaryR=2;Wavenumber,k=20.

;q

InallthecomputationsbelowtheconvergenceofthegeneralizedFEMsolutionuph,willbeobtainedbyincreasingpandqontheCartesianmeshesshowninFig.5.

5.1.p-andq-convergenceofthegeneralizedFEM

Inalltheresultsbelow,unlessexplicitlystated,theerrorduetoevaluationofKandFbynumericalintegrationisneg-ligible.Hereweemployedthe40·40Gauss–Legendreintegrationruleinthesquaresnotintersectingtheboundary,whileinthesquareswhichintersecttheboundaryweemployedthe13·13Gauss–Legendreruleinthesubelementsobtainedaftersubdividingtheelementsuniformly9timesandusingblendingfunctiontransformationinthesubelementsadjacenttotheboundary.Tocomputetheloadvectorandstiffnessmatrixcontributionsoftheboundarytermsweused,piecewise,the40pointGauss–Legendreintegrationrule.Thesechoicespracticallyeliminatethenumericalintegrationerrorforka=20,andfortherangeofaccuracysoughtinthecomputations.

;0

InTable1wereportthepercentrelativeerrorkrðuÀuphÞkL2ðXÞ=krukL2ðXÞÂ100%,thecorrespondingpercentrelativep;q;Ã;q;ÃerrorinthebestapproximationkrðuÀAhuÞkL2ðXÞ=krukL2ðXÞÂ100%,whereApudenotesthebestapproximationofuh

whichwascomputedusingnumericalintegrationandisdefinedbelow,andthevalueoftheirratio

;0p;q;Ã

krðuÀupuÞkL2ðXÞwhichmeasuresthepollution,forMeshA,B,andC,andforp=1,...,5.WewillhÞkL2ðXÞ=krðuÀAh

1saythatthepollutionintheH-seminormoftheerrorisnegligibleif

;0

krðuÀuphÞkL2ðXÞ;q;ÃkrðuÀApuÞkL2ðXÞh

À1<󰀂ð34Þ

(a)(b)(c)Fig.5.TheCartesianmeshesusedinthecomputations:(a)MeshA,h=0.75;(b)MeshB,h=0.375;(c)MeshC,h=0.1875.Table1

;0

ThevaluesofthepercentrelativeerrorintheGFEMsolution:krðuÀuphÞkL2ðXÞ=krukL2ðXÞÂ100%(firstline),thebestapproximation:

;q;Ã;0p;q;Ã

uÞkL2ðXÞ=krukL2ðXÞÂ100%(secondline),andtheirratiokrðuÀupuÞkL2ðXÞ(thirdlineinparenthesis)krðuÀAphhÞkL2ðXÞ=krðuÀAh

p=1

MeshA

h=0.75MeshBh=0.375MeshCh=0.1875

99.9009.1295(1.0078)127.288696.2143(1.3230)185.395581.0208(2.2883)

p=2101.829196.8871(1.0510)160.254683.4217(1.9210)109.186736.7081(2.9746)

p=3128.9875.3488(1.4437)120.556354.4879(2.2126)14.825011.4461(1.2952)

p=4117.288474.9621(1.57)46.441627.0861(1.7146)2.73812.6833(1.0204)

p=5103.272456.8233(1.8174)12.699910.6751(1.17)0.50460.5047(0.9980)

T.Strouboulisetal./Comput.MethodsAppl.Mech.Engrg.195(2006)4711–47314719

forsufficientlysmall󰀂>0.Intheresultsbelowweseethatwhentheerrorissmall,theratiokrðuÀuphÞkL2ðXÞ=p;q;Ã

krðuÀAhuÞkL2ðXÞmaybebelowone.Thisisduetotheerrorsintheemployednumericalintegrationsoftheright-hand

p;q;Ã

sideofthediscreteproblemforAhu,andthisonlyhappensforrathersmallerrorandwhenthepollutionisnegligible.Remark7.LetusalsonotethatforrectangularelementsitisalsopossibletoevaluateallintegralsanalyticallyfollowingBettessetal.[28,47].Neverthelessthismaynotbethecaseforelementsintersectingtheboundarywhereweneedtoemploynumericalintegration.

;q;Ã;q

LetusnowelaborateonthecomputationofthebestapproximationApu.LetAphhubethebestapproximationofuintheH1-seminormdefinedbythediscretevariationalproblem,

;qk;q

Bestapproximation:FindAphu2Sh;psuchthatZZ

p;q;q

rðAhuÞÁr󰀂vdX¼ruÁr󰀂vdX8v2Skð35Þh;p;

X

X

whereuisgivenby(31).Thisleadstothesystemoflinearequations

KBAUBA¼FBA;

ð36Þ

whereKBAconsistsoftheLaplacianpartofKusedinthediscretesystemofHelmholtzGFEMequations.Theright-hand

sideFBAisnotcomputedwhensettingupthediscreteHelmholtzGFEMequations,anditsevaluationrequiresthenumer-icalintegrationofintegralsofthetype

ZZ󰀊󰀋

ðiÞDh

ruÁrNkdX;ruÁr/iWjdX;ð37Þ

X

X

h

respectively,fortheFEdegreesoffreedom,andtheplane-wavedegreesoffreedomineachpatchxDi.Thesenumericalintegrationerrorsfortheemployednumericalquadratureintheevaluationofthesetermsarenotnegligibleandarerespon-;q

siblefortheparadoxicalbehavioroftheH1-seminormoftheerrorintheHelmholtzGFEMsolutionuphbeingsmallerthatp;q;ÃthecorrespondingvalueoftheH1-seminormoftheerrorinthebestapproximationAhu,whentherelativeerrorisvery

p;qp;qp;q;Ã

small,whenuhandAhushouldbepracticallyidentical.ForthisreasonweemployedthestarinAhutounderlinetheeffectofnumericalintegration.

;q

InTheorem2abovewehavestatedthatforsufficientlysmall(1+k2)h,theHelmholtzGFEMsolutionuphconverges;qlikethebestapproximationAphu,namelythereexistsaCsuchthat

;qp;q

krðuÀuphÞkL2ðXÞ6CkrðuÀAhuÞkL2ðXÞ.

ð38Þ

Nevertheless,thepresenttheorydoesnotindicatewhatissufficientlysmall(1+k2)h.Wewillnowgivesomeresultswhichindicatewhen(1+k2)hissufficientlysmallinthesettingofourmodelexample.

p;0

InTable1wegivethevaluesofthepercentrelativeerrorintheGFEMsolution:krðuÀuhÞkL2ðXÞ=krukL2ðXÞÂ100%,

p;q;Ãp;0p;q;Ã

thebestapproximation:krðuÀAhuÞkL2ðXÞ=krukL2ðXÞÂ100%,andtheirratiokrðuÀuhÞkL2ðXÞ=krðuÀAhuÞkL2ðXÞ

2forMeshA,B,andC,forp=1,...,5,andq=0.Wecanseethatinthepreasymptoticrange,when(1+k)hisnotsuf-;0

ficientlysmall,theerrorcangrowwithdecreasinghandincreasingp;forexampletheerrorkrðuÀuphÞkL2ðXÞgrowswhenhishalvedfromMeshAtoMeshBforp=1,2,andforMeshA,aspisincreasedfromp=1top=3,andforMeshBfor

p;q;Ã

p=1top=2.Ontheotherhand,theerrorinthebestapproximationkrðuÀAhuÞkL2ðXÞdecreasesmonotonicallywhenhishalvedandpisincreased.

FromTable1itisclearlyvisiblethatthepollutioneffectissmallerforhigherorderelements,whichisincompleteagree-ˇka[37].Itisclearthatthegenericstatement‘‘for(1+k2)hsufficientlysmall’’mentwiththeanalysisofIhlenburgandBabus

isnotpreciseenoughandmoreworkisneededforcomingupwithmorepreciseassumptions.

Letusnowconsiderthecaseoffixedhwheretheconvergenceisobtainedbyincreasingpandq.Table2givesthepercent

;q

relativeerrorintheGFEMsolutionkrðuÀuphÞkL2ðXÞ=krukL2ðXÞÂ100%forp=1,...,5,andq=1,...,17,thecorre-p;q;ÃspondingpercentrelativeerrorinthebestapproximationkrðuÀAhuÞkL2ðXÞ=krukL2ðXÞÂ100%andthevalueoftheir

p;qp;q;Ã

ratiokrðuÀuhÞkL2ðXÞ=krðuÀAhuÞkL2ðXÞforMeshA.

NotethatinMeshAforp=1(resp.p=2)thepollutionissignificantuptoq=17(resp.q=11)andisresponsibleforthenon-monotonicconvergenceoftheGFEMsolution.ThisisbecausehofMeshAisnot‘‘sufficientlysmall’’.HoweverforpP3,thesamehissufficientlysmallandtheasymptoticcharacteristicsareachievedforqP9forp=3andqP7for

;qp;q

p=4,5,andtheerrorintheGFEMHelmholtzsolutionuphisveryclosetotheerrorinthebestapproximationAhu.ThisisvisibleinFigs.6and7wherewegivetheconvergenceoftherelativeerrorintheGFEMbestapproximationandtherelativeerrorintheHelmholtzGFEMsolutionversusqforp=1andp=5.Wecanclearlyseethatwhathissufficientlysmalldependsontheemployedp.

Fig.8givestheq-convergenceoftheGFEMbestapproximationandtheGFEMHelmholtzsolutionforMeshAforpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip=1,...,5andq=0,1,...,17.HereweemployedthesquarerootofthetotalnumberofdegreesoffreedomNDOFinthehorizontalaxis.Wenotethatwepracticallyhaveexponentialconvergencewith

4720T.Strouboulisetal./Comput.MethodsAppl.Mech.Engrg.195(2006)4711–4731

Table2

;q

ThevaluesofthepercentrelativeerrorintheGFEMsolution:krðuÀuphÞkL2ðXÞ=krukL2ðXÞÂ100%(firstline),thebestapproximation:

;q;Ã;qp;q;Ã

krðuÀApuÞkL2ðXÞ=krukL2ðXÞÂ100%(secondline),andtheirratio:krðuÀupuÞkL2ðXÞ(thirdlineinparenthesis)forMeshAhhÞkL2ðXÞ=krðuÀAhp1

q=188.56

67.7462(1.3122)92.785165.8201(1.4097)86.494359.6792(1.4493)96.574650.0543(1.9294)71.191338.6725(1.8409)

q=387.152751.6927(1.6860)87.753849.9326(1.7574)88.333745.1638(1.9558)80.086237.5849(2.1308)53.7927.3563(1.9663)

q=575.090929.4483(2.5500)73.631428.5353(2.5804)72.002426.0802(2.7608)45.335821.0737(2.1513)24.659813.6805(1.8026)

q=724.09106.1581(3.9121)22.16185.8758(3.7717)73.28514.8816(15.0120)4.40483.6280(1.2141)2.68192.4309(1.1033)

q=913.49842.2310(6.0502)3.84572.0874(1.8423)1.93491.7118(1.1303)1.40731.3216(1.08)0.90410.8691(1.0403)

q=112.00351.0542(1.9005)6.04320.9945(6.0766)0.83950.7800(1.0763)0.51130.4934(1.0363)0.31220.3020(1.0338)

q=134.88090.4320(11.2980)0.45360.3936(1.1524)0.34100.3213(1.0613)0.22110.2137(1.0346)0.14090.1376(1.0240)

q=150.28000.1329(2.1068)0.13340.1226(1.0881)0.10200.0981(1.0398)0.07580.0740(1.0243)0.04660.0460(1.0130)

q=170.06240.0454(1.3744)0.05060.0431(1.1740)0.03590.0351(1.0228)0.02670.02(1.0114)0.01710.0176(0.9716)

2

3

4

5

GFEMbest approximationRelative error in H1 seminorm (in percent) 100Helmholtz 10Best Approximation1 Mesh Ah = 0.75 0.1h 0.01 0 10 20pollution ends 30 40 50NDOFFig.6.q-convergenceoftheGFEMandbestapproximationforMeshA,forp=1. 1000GFEMbest approximationRelative error in H1 seminorm (in percent) 100Helmholtz 10Best ApproximationMesh Ah = 0.75pollution ends1 0.1 0.01 28h 30 32 34 36 38 40 42 44 46 48 50NDOFFig.7.q-convergenceoftheGFEMandbestapproximationforMeshA,forp=5.T.Strouboulisetal./Comput.MethodsAppl.Mech.Engrg.195(2006)4711–4731

10004721

Relative error in H1 seminorm (in percent) 100 10GFEM, p = 1BA, p = 1GFEM, p = 2BA, p = 2GFEM, p = 3BA, p = 3GFEM, p = 4BA, p = 4GFEM, p = 5BA, p = 5Mesh Ah = 0.751 21 0.1h 0.01 0 10 20 30 40 50NDOF;qp;q;ÃFig.8.q-convergenceoftheGFEMsolutionupuforMeshA,forp=1,...,5andhversusthecorrespondingconvergenceofthebestapproximationAhq=0,1,...,17.Notethedependenceofthepre-asymptoticrangeonp.;qÀc

krðuÀupDhÞk%Ce

pffiffiffiffiffiffiffiffiffiffi

NDOF

ð39Þ

withc%0.5.ThisfollowsfromTheorems2and3whichstatethat,forsufficientlysmallh,wehaveexponentialconver-pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

gencewithq.NotethatNDOFisproportionalwithq.

;q

InTable3wegivethepercentrelativeerrorintheGFEMsolution:krðuÀuphÞkL2ðXÞ=krukL2ðXÞÂ100%,thebest

;q;Ãp;qp;q;Ã

approximation:krðuÀApuÞkL2ðXÞ=krukL2ðXÞÂ100%,andtheirratio:krðuÀuhÞkL2ðXÞ=krðuÀAhuÞkL2ðXÞforMeshh

B,whereweobtainedtheconvergencebyincreasingpandq.Wenoteherethatthepollutionismuchsmallercompared

;q;Ã

withthesimilarpandqonMeshA.Notealsothatforp=4,5,andq=11,13,thebestapproximationApuispollutedh

bytheerrorinthenumericalintegrationofitsright-handsideandwearegettingtheparadoxicalbehaviorthatwemen-;qp;q

tionedearlier.Thishappenswellafterthepollutionhasendedandwhenwepracticallyhaveuph%Ahu.ThiscanbeclearlyseeninFig.9.

InFig.10,weshowtheq-convergenceforp=1,...,5onMeshC.Theresultsherearepracticallyexactforq=7andq=9.ComparingFigs.9and10weseethedependenceoftheexponentialrateofconvergenceconh.

FromtheaboveresultsweseethatthemosteffectivewaytoemploythegeneralizedFEMfortheHelmholtzistouseacoarsemesh,withh=ak=a2p/k,witha2(1,2.5)e.g.about1or2wavelengthperelement,tofixp=2or3,andtoobtainconvergencebyincreasingq.Thisisanalogouswiththep-convergenceemployedbythecommercialcodeStressCheckandotherp-versioncodes.

Table3

;q

ThevaluesofthepercentrelativeerrorintheGFEMsolution:krðuÀuphÞkL2ðXÞ=krukL2ðXÞÂ100%(firstline),thebestapproximation:

;q;Ã;qp;q;Ã

krðuÀApuÞkL2ðXÞ=krukL2ðXÞÂ100%(secondline),andtheirratio:krðuÀupuÞkL2ðXÞ(thirdlineinparenthesis)forMeshBhhÞkL2ðXÞ=krðuÀAhp1

q=0127.2886

96.2143(1.3230)160.254683.4217(1.9210)120.556354.4879(2.2125)46.441627.0861(1.7146)12.699910.6751(1.17)

q=196.8660.7625(1.5942)88.076350.2974(1.7511)67.515729.9357(2.2554)24.347513.8615(1.7565)6.27525.1855(1.2101)

q=3.322722.7306(2.8298)44.722519.4872(2.2950)22.594912.2103(1.8505)7.55915.5327(1.3663)2.28762.0518(1.1149)

q=54.38633.4416(1.2745)3.09112.8093(1.1003)1.72181.6559(1.0398)0.73930.7305(1.0120)0.24980.2524(0.97)

q=70.86690.8270(1.0482)0.60430.5916(1.0215)0.30440.3018(1.0086)0.11290.1124(1.0044)0.04160.0421(0.9881)

q=90.13940.1381(1.0094)0.10480.1041(1.0067)0.06250.0622(1.0048)0.02270.0230(0.9870)0.00870.0094(0.9255)

q=110.02150.0214(1.0047)0.01740.0174(1.0000)0.01080.0108(1.0000)0.00190.0053(0.3585)0.00190.0049(0.3878)

q=130.00320.0033(0.9697)0.00290.0030(0.9667)0.00190.0024(0.7917)0.00090.0043(0.2093)0.00040.0144(0.0278)

2

3

4

5

4722T.Strouboulisetal./Comput.MethodsAppl.Mech.Engrg.195(2006)4711–4731

1000Relative error in H1 seminorm (in percent) 100 10GFEM, p = 1BA, p = 1GFEM, p = 2BA, p = 2GFEM, p = 3BA, p = 3GFEM, p = 4BA, p = 4GFEM, p = 5BA, p = 5 12Mesh Bh = 0.3751 0.1 0.01 0.001h 0.0001 10 20 30 40 50 60 70 80NDOF;qp;q;ÃFig.9.q-convergenceoftheGFEMsolutionupuforMeshB,forp=1,...,5andhversusthecorrespondingconvergenceofthebestapproximationAhq=0,1,...,13.Notetheparadoxicalbehaviorofthebestapproximationwhichisduetonumericalintegrationerrorintheright-handsideofthediscreteequations. 1000 100 10 1Relative error in H1 seminorm (in percent)q-conv, p=1q-conv, p=2q-conv, p=3q-conv, p=4q-conv, p=54 0.1 0.01 0.001 0.0001Mesh Ch = 0.18751h 1e-005 0 20 40 60 80 100 120 140NDOF;qFig.10.q-convergenceoftheGFEMsolutionuphforMeshC,forp=1,...,5,andq=0,1,...,9.ComparingthisfigurewithFig.9weseethedependenceoftheexponentialrateofconvergencecwithh.Aboveweansweredwhatissufficientlysmallhinthesettingofourmodelexample.Now,comparingTables2and3wealsoseetheeffectofthenumericalintegrationontheconvergenceofthebestapproximationwithhwhichwasrecently

ˇkaetal.in[45].addressedbyBabus

FromTheorems2and3,weexpectthatforsufficientlysmallh,wegetexponentialconvergencewithpandq,namely

;qÀðaqþbpÞ

krðuÀuphÞkL2ðXÞ%Ce

ð40Þ

withC;a,andbdependingonh.LetusnowanalyzethecomputedresultsandaddresstheexponentialratesoftheGFEMinourmodelexample.Figs.11–13graphtheconvergenceoftherelativeerrorversusqforMeshA,B,andC,respectively.Asexpected,forsufficientlysmallh,wegetalinearrelationshipwithpandq,namely

! !p;q

krðuÀuhÞkL2ðXÞClog%logÀðaqþbpÞ.ð41Þ

krukL2ðXÞkrukL2ðXÞ

T.Strouboulisetal./Comput.MethodsAppl.Mech.Engrg.195(2006)4711–4731

10004723

Relative error in H1 seminorm (in percent) 100q-conv, p=1q-conv, p=2q-conv, p=3q-conv, p=4q-conv, p=5 10 1Mesh Ah = 0.75 0.11h2 8 10 12 14 16 18 0.01 0 2 4 6q;qFig.11.q-convergenceoftherelativeerroroftheGFEMsolutionuphforMeshAversusq,forp=1,...,5. 1000Relative error in H1 seminorm (in percent) 100q-conv, p=1q-conv, p=2q-conv, p=3q-conv, p=4q-conv, p=5 10143 0.1Mesh Bh = 0.375 0.01 0.001 0.0001h02468 10 12 14q;qFig.12.q-convergenceoftherelativeerroroftheGFEMsolutionuphforMeshBversusq,forp=1,...,5. 1000 100Relative error in H1 seminorm (in percent)q-conv, p=1q-conv, p=2q-conv, p=3q-conv, p=4q-conv, p=5 101 0.1 0.01 0.001 0.0001 1e-005 1e-00611Mesh Ch = 0.1875h02468 10q;qFig.13.q-convergenceoftherelativeerroroftheGFEMsolutionuphforMeshCversusq,forp=1,...,5.4724T.Strouboulisetal./Comput.MethodsAppl.Mech.Engrg.195(2006)4711–4731

InFig.11,weseeoscillationswhichindicatepre-asymptoticbehaviour,whilemostoftheresultsinMeshBforqP5,andMeshCforqP3areintheasymptoticrange.Comparingasymptoticrangeinthesefiguresitisclearthattherateofexponentialconvergenceincreasesashisrefined,forexampleforMeshAwehavea%0.5,forMeshB,a%0.75,andforMeshC,a%1.Theimprovementintheexponentialratesaashisdecreasedcanbeexplainedbynotingthat,asthemeshisrefined,thesolutionbecomessmootherlocallywithrespecttothemesh.

Letusoncemoreunderlinethatherewearemainlyinterestedintheexponentialconvergencewithq,andweuseptolimittheextentofthepreasymptoticrange.5.2.Aposterioriestimationusingextrapolation

Aswehaveseenabove,amainopenprobleminthesolutionoftheHelmholtzequationbythep-versionofthegeneral-izedFEMiswhenthepollutionends,orequivalentlywhatissufficientlysmallh.Althoughatthismomentwecannotanswerthisquestiontheoretically,wemaydetecttheendofthepollutioninasimpleaposterioriway.Further,havingdetectedtheendofthepollution,intheasymptoticrangewecanemployextrapolationtoobtainreliableerrorestimators.Letusgiveanexampleofhowwecandetecttheendofthepollutionandhowwecanuseq-extrapolationforestimating

;q

theerrorinacomputedquantityofinterestFðuphÞbyextrapolation.Letusunderlinethatbeforeapplyinganyextrapo-lationweneedtomakesurethatthevaluesemployedintheapproximationareintheasymptoticrange.Thiscanbedeter-pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip;q

minedbyemployingagraphofthevalueofthecomputedoutputFðuhÞversusqorequivalentlyversusNDOF.Letus,forexample,employtheH1-seminormofthesolutionasthequantityofinterest,namelyFðuÞ¼krukL2ðXÞ.pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip;qp;q

InFig.14(resp.Fig.15)weplottedFðuhÞ¼kruhkL2ðXÞversusNDOFforp=1,...,5,andq=0,1,...,17,forMeshA(resp.q=0,1,...,13forMeshB).Frombothfiguresitisrathereasytodetecttheendofthepollutionandtheextentoftheasymptoticrangeforwhichweexpecttheextrapolationtogivegoodresults.Wecannowemploythesimpleextrapolationproposedin(28)abovetoobtaintheestimator

p;qEskruhkL2ðXÞ

¼

p;qþs;q

kruhkL2ðXÞÀkruphkL2ðXÞ

p;qþskruhkL2ðXÞ

ð42Þ

;q

fortherelativeerrorinkruphkL2ðXÞ.Weseethatbyemployingq=11,s=6forMeshA,andq=7,s=6forMeshB,wegeteffectivitiesbetween0.67and1.67(seeTables4and5).Itisalsopossibletoemployamoresophisticatedextrapolation

´[42,p.69].ˇkaandSzabotoextracthigherorderaccuracy,as,e.g.inBabus

Letusunderlinethatthisq-extrapolationapproachcanbeusedfortheaposterioriestimationoftheerrorinanyquan-tityofinterestanddoesnotrequireanyextracoding.

Comparingtheresultsofouraposterioriapproachfordetectingtheendofthepollutionandforestimatingtheerror,weseethatwegetarobustmethodwithrespecttothemeshsizeh.Weseethatthecharacterofthegraphsandtheeffectivityindicesarepracticallythesameforbothmeshes.

50 45H1 seminorm of the approximate solution 40 35 30 25 20 15 105q-conv, p=1q-conv, p=2q-conv, p=3q-conv, p=4q-conv, p=5asymptotic behavior}Mesh Ah = 0.75pollution ends(u) = ||∇u||2Lh05 10 15 20 25 30 35 40 45 50NDOFpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi;qp;qFig.14.GraphoftheoutputFðupNDOFforMeshA.FromthisgraphwecandetectwherethepollutionhaspracticallyendedhÞ¼kruhkL2ðXÞversusandtheasymptoticrangeforwhichwecanemploytheextrapolationwithgoodresults.T.Strouboulisetal./Comput.MethodsAppl.Mech.Engrg.195(2006)4711–4731

50 484725

H1 seminorm of the approximate solution 46 44 42 40(u) = ||∇u||L2q-conv, p=1q-conv, p=2q-conv, p=3q-conv, p=4q-conv, p=5}asymptotic behaviorpollution endsh 30 40 50 60 70 80 38 36 34 32 30 28 10Mesh Bh = 0.375 20NDOF;qp;qFig.15.GraphoftheoutputFðuphÞ¼kruhkL2ðXÞversuspffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNDOFforMeshB.Table4

;11

ExactandestimatedrelativeerroranditseffectivityindexfortherelativeerrorinkruphkL2ðXÞ,computedonMeshAp

p;11

krukL2ðXÞÀkruhkL2ðXÞ

krukL2ðXÞ

1

2345

0.03860.17530.01050.00090.0008

(%)

;17p;11

kruphkL2ðXÞÀkruhkL2ðXÞ

;17kruphkL2ðXÞ

(%)h0.9819

0.99711.00000.66671.2500

0.03790.17480.01050.00060.0010

Table5

;7

ExactandestimatedrelativeerroranditseffectivityindexfortherelativeerrorinkruphkL2ðXÞ,computedonMeshBp

;7

krukL2ðXÞÀkruphkL2ðXÞ

krukL2ðXÞ

1

2345

0.00850.00570.00050.00010.00003

(%)

p;13;7kruhkL2ðXÞÀkruphkL2ðXÞ

;13kruphkL2ðXÞ

(%)h0.9882

0.98251.60001.00001.6667

0.00840.00560.00080.00010.00005

5.3.Theeffectsofperturbationofthemesh

Theoretically,perturbationofthemeshcouldinfluenceboththeapproximityandstabilityofthemethod,throughtheconstantCinTheorem3.Togettheideahowmuchthisinfluencecanbeweemployedacoupleofsamplecomputations.

;q

TocheckthesensitivityoftheGFEMsolutionoftheHelmholtzproblemuphwithrespecttoperturbationsofthemesh,weemployedtheperturbedMeshBandMeshCshowninFig.16,andinFigs.17and18wecomparetheconvergenceoftherelativeerrorfortheunperturbedandperturbedmeshes.FromthesegraphsitisclearthattheconvergenceoftheGFEMsolutionisnotsensitivewithrespecttoperturbationsofthenodesofthemesh.5.4.Theeffectsofroundofferror

Thetheoreticalperformanceofthemethodwasestablishedundertheassumptionthatthereisneitherroundoffnorquadratureerror.Nextwewantedtogetanideaofhowmuchtheroundofferrorcaninfluencetheresults.

;q

TostudytheeffectoftheroundofferrorintheconvergenceofthegeneralizedFEMsolutionuph,weemployedasourmodelexample,theHelmholtzequationontheunitsquareX=(0,1)·(0,1)withRobinboundaryconditions

T.Strouboulisetal./Comput.MethodsAppl.Mech.Engrg.195(2006)4711–4731

Fig.16.PerturbedMeshBandMeshC. 1000q-conv, p=1, pertq-conv, p=1)tq-conv, p=2, pertq-conv, p=2n 100ecq-conv, p=3, pertq-conv, p=3rep n 10hq-conv, p=4, pertq-conv, p=4i(q-conv, p=5, pertq-conv, p=5 mronim 1es2 1H 0.1Mesh Bnih = 0.3751 rorre 0.01evitaleR 0.001 0.0001 10 20 30 40 50 60 70 80NDOFFig.17.ConvergencecurvesforperturbedandunperturbedMeshB. 1000q-conv, p=1q-conv, p=1, pert 100q-conv, p=2)tq-conv, p=2, pertneq-conv, p=3crq-conv, p=3, pertep 10q-conv, p=4 nhq-conv, p=4, perti( q-conv, p=5mq-conv, p=5, pertr 1onimes 0.1 14HMesh C ni 0.01h = 0.18751rorre ev 0.001italeR 0.0001 1e-005 0 20 40 60 80 100 120 140NDOFFig.18.ConvergencecurvesforperturbedandunperturbedMeshC.ÀDuþk2u¼0inX;ou

on

þiku¼gonoX;ð43aÞð43bÞ

4726T.Strouboulisetal./Comput.MethodsAppl.Mech.Engrg.195(2006)4711–47314727

wherewechosegsuchthattheexactsolutionistheplanewave

p

uðx;yÞ¼eikðxcoshþysinhÞ;h¼.

16

ð44Þ

ForthisexampletherealreadyexistresultsbyMelenk[3]forp=0,q=1,...,19onuniformN·NmeshesofsquaresforN=1,2,4,and8.

;q

Tables6–8comparetheresultsfortherelativeerrorkrðuÀuphÞkL2ðXÞ=krukL2ðXÞcomputedusingp=1andq=1,3,...,19,onthe4·4meshfork=1,2,4,8,16,32,and,usingdoubleprecisionintheWindowsmachine(Table6),andtheLinuxmachine(Table7),withtheresultscomputedusingquadrupleprecisionintheUnixmachine(Table8).TheresultsbyMelenkareavailableinhisPh.D.thesisandisincludedinthegraphsbelow.

Figs.19–22givetheconvergencegraphsfork=1,16,32,and.Notethatforlowktheconvergenceislimitedbythemachineprecision.Thisisbecausetheemployedwavefunctionsarelinearlydependentatthelimitofzerowavenumberk=0.Notealsothatasthewavenumberkisincreasedthereissmalldifferenceduetothemachineprecision.Forexample,fork=,allresultspracticallycoincideandtheaccuracyoftheapproximationislimitednotbytheroundoffbutbythenumericalintegrationerror.Nevertheless,forexactintegration,theexponentialconvergencecanalwaysbemaintainedto

Table6

Convergenceusing4·4mesh,usingdoubleprecisioninWindowsmachine.Theresultsbelowthehorizontallinesarepollutedbytheroundoffindoubleprecisionq135791113151719

k=1.00.2750DÀ020.1900DÀ060.58DÀ070.1507DÀ060.2983DÀ060.1650DÀ060.3206DÀ060.3572DÀ050.1673DÀ060.1719DÀ06

k=2.00.5519DÀ020.2424DÀ050.8562DÀ070.5798DÀ060.7242DÀ060.3113DÀ060.8369DÀ060.8628DÀ060.3871DÀ060.9188DÀ06

k=4.00.1121DÀ010.7345DÀ040.55DÀ070.8622DÀ070.3922DÀ060.1344DÀ060.2596DÀ060.53DÀ060.1636DÀ060.4349DÀ06

k=8.00.2363DÀ010.1811DÀ020.1982DÀ040.2592DÀ070.5706DÀ070.1780DÀ060.1302DÀ060.1850DÀ060.1995DÀ060.23DÀ06

k=16.00.59DÀ010.1768DÀ010.2368DÀ020.5621DÀ040.91DÀ060.8442DÀ080.4207DÀ070.2279DÀ060.5962DÀ060.1175DÀ06

k=32.00.1820D+000.5667D+000.6921DÀ010.3792DÀ010.8337DÀ030.7908DÀ040.1426DÀ050.3379DÀ080.4638DÀ070.1766DÀ06

k=.00.6907D+000.7042D+000.7317D+000.8227D+000.1639D+000.4165DÀ010.4636DÀ020.1371DÀ030.2173DÀ040.1747DÀ04

Table7

Convergenceusing4·4mesh,usingdoubleprecisioninLinuxmachineq135791113151719

k=1.00.2750DÀ020.9192DÀ070.9855DÀ070.9068DÀ070.4880DÀ060.6618DÀ060.1237DÀ060.1913DÀ060.6194DÀ060.3583DÀ06

k=2.00.5519DÀ020.2424DÀ050.5841DÀ070.4599DÀ060.4840DÀ060.5812DÀ060.6510DÀ060.7104DÀ060.4544DÀ060.3831DÀ06

k=4.00.1121DÀ010.7345DÀ040.55DÀ070.2425DÀ060.11DÀ060.2132DÀ060.2399DÀ050.2045DÀ060.13DÀ060.4550DÀ06

k=8.00.2363DÀ010.1811DÀ020.1982DÀ040.2591DÀ070.1195DÀ060.1157DÀ060.1315DÀ060.2281DÀ060.3082DÀ060.3862DÀ06

k=16.00.59DÀ010.1768DÀ010.2368DÀ020.5621DÀ040.91DÀ060.26DÀ070.1441DÀ060.1318DÀ060.17DÀ060.1261DÀ06

k=32.00.1820D+000.5667D+000.6921DÀ010.3792DÀ010.8337DÀ030.7908DÀ040.1426DÀ050.3476DÀ080.5919DÀ070.7770DÀ07

k=.00.6907D+000.7042D+000.7317D+000.8227D+000.1639D+000.4165DÀ010.4636DÀ020.1371DÀ030.2173DÀ040.1747DÀ04

Table8

Convergenceusing4·4mesh,usingquadrupleprecisioninUnixmachineq135791113151719

k=1.00.2750EÀ020.7666EÀ070.2320EÀ120.2340EÀ150.3914EÀ150.3011EÀ140.66EÀ140.1993EÀ140.1251EÀ140.6050EÀ14

k=2.00.5519EÀ020.2424EÀ050.1168EÀ090.5970EÀ150.1656EÀ140.3018EÀ140.3552EÀ150.8273EÀ150.1202EÀ140.8296EÀ14

k=4.00.1121EÀ010.7345EÀ040.5586EÀ070.4509EÀ110.9733EÀ160.3061EÀ150.4621EÀ150.2594EÀ140.1277EÀ140.7191EÀ15

k=8.00.2363EÀ010.1811EÀ020.1982EÀ040.2583EÀ070.1103EÀ100.1634EÀ140.1905EÀ150.3105EÀ150.1135EÀ140.1269EÀ14

k=16.00.59EÀ010.1768EÀ010.2368EÀ020.5621EÀ040.91EÀ060.1542EÀ080.8322EÀ120.7876EÀ160.73EÀ160.1403EÀ15

k=32.00.1820E+000.5667E+000.6921EÀ010.3792EÀ010.8337EÀ030.7908EÀ040.1426EÀ050.3223EÀ080.4672EÀ100.1828EÀ11

k=.00.6907E+000.7042E+000.7317E+000.8227E+000.1639E+000.4165EÀ010.4636EÀ020.1371EÀ030.2173EÀ040.1747EÀ04

4728T.Strouboulisetal./Comput.MethodsAppl.Mech.Engrg.195(2006)4711–4731

1001 0.01 0.0001 1e-006 1e-008 1e-010 1e-012 1e-014 1e-0161x1 mesh, windows1x1 mesh, Linux1x1 mesh, Babuška1x1 mesh, quadruple2x2 mesh, windows2x2 mesh, Linux2x2 mesh, Babuška2x2 mesh, quadruple4x4 mesh, windows4x4 mesh, Linux4x4 mesh, Babuška4x4 mesh, quadruple8x8 mesh, windows8x8 mesh, Linux8x8 mesh, Babuška 40 50 60Relative error in H1 seminorm (in percent)quadruple precision0 10 20 30NDOFˇka,Windowsmachine,Linuxmachine,andquadrupleprecisionfork=1.HerewelabelonlythegraphsFig.19.ComparisonofresultsofBabuscomputedusingquadrupleprecision. 1001 0.01 0.0001 1e-006 1e-008 1e-010 1e-012 1e-014 1e-0161x1 mesh, windows1x1 mesh, Linux1x1 mesh, Babuškaquadruple1x1 mesh, quadrupleprecision2x2 mesh, windows2x2 mesh, Linux2x2 mesh, Babuška2x2 mesh, quadruple4x4 mesh, windows4x4 mesh, Linux4x4 mesh, Babuška4x4 mesh, quadruple8x8 mesh, windows8x8 mesh, Linux8x8 mesh, Babuška0 10 20 30 40 50 60double precisionRelative error in H1 seminorm (in percent)NDOFˇka,Windowsmachine,Linuxmachine,andquadrupleprecisionfork=16.Fig.20.ComparisonofresultsofBabussmallerrelativeerrorbyquadrupleprecision.Similarly,ifwehadthecapabilitytoemployoctupleprecisionwecouldmain-taintheexponentialconvergencetoevenhigheraccuracy.Alldoubleprecisioncomputations(Linux,Windows,Melenk)

givepracticallyidenticalresultsuntiltheroundofferrorbecomessignificant.

Letus,oncemore,underlinethatthedifferenceinthecomputedresultsareduetotheroundoffeffectintheemployeddirectsolver.Itisalsointerestingtonotethatthevariousdoubleprecisioncomputationgivedifferentresults,thedifferencebecomesvisiblewhenwereachthelimitofaccuracybelowthehorizontallineinthecolumns.

Remark8.LetusalsonotethattheconditioningofthepartitionofunitymethodusingsystemsofplanewaveshasbeenaddressedinthepaperbyLaghrouche,Bettess,andAstley,whereitwasconcludedthat‘‘itispossibletodetermine‘safe’regionsintheqÀkdomainforwhichtheconditionnumberstayswithinacceptablelimitsonanelementbyelementbasis...’’.Formoredetailssee[48].

Remark9.HereandinotherrelatedworksthegeneralizedFEM/PartitionofUnityMethodwasbasedonusingplane-wavesasthelocalbasis.Howeveritisanopenquestionwhatistheoptimalchoiceoflocalbasis;seealso[49]wherewegivesamplecomparisonsofthePartitionofUnityMethodusingplane-wavesversusthesamemethodusingwave-bands.Ofcoursemanyotherchoicesarepossible,like,forexamplethegeneralizedharmonicpolynomialsofVekuaproposedin[3],etc.

T.Strouboulisetal./Comput.MethodsAppl.Mech.Engrg.195(2006)4711–4731

100001x1 mesh, windows1x1 mesh, Linux1x1 mesh, Babuška1x1 mesh, quadruple2x2 mesh, windows2x2 mesh, Linux2x2 mesh, Babuška2x2 mesh, quadruple4x4 mesh, windows4x4 mesh, Linux4x4 mesh, Babuška4x4 mesh, quadruple8x8 mesh, windows8x8 mesh, Linux8x8 mesh, Babuška4729

Relative error in H1 seminorm (in percent) 1001 0.01 0.0001 1e-006 1e-008 1e-0100 10 20 30 40 50 60NDOFˇka,Windowsmachine,Linuxmachine,andquadrupleprecisionfork=32.NotethatthereispracticallyonlyFig.21.ComparisonofresultsofBabussmalldifferencebetweentheresultscomputedindoubleandquadrupleprecision. 1000 100 101 0.1 0.01 0.001 0.0001 1e-005 1e-00601x1 mesh, windows1x1 mesh, Linux1x1 mesh, Babuška1x1 mesh, quadruple2x2 mesh, windows2x2 mesh, Linux2x2 mesh, Babuška2x2 mesh, quadruple4x4 mesh, windows4x4 mesh, Linux4x4 mesh, Babuška4x4 mesh, quadruple8x8 mesh, windows8x8 mesh, Linux8x8 mesh, BabuškaRelative error in H1 seminorm (in percent)integration error 10 20 30 40 50 60 70NDOFˇka,Windowsmachine,Linuxmachine,andquadrupleprecisionfork=.NotethattheresultsfortheFig.22.ComparisonofresultsofBabuscomputationindoubleandquadrupleprecisionpracticallycoincides.6.Conclusionsandopenproblems

WehavedevelopedtheGFEMfortheHelmholtzequationforhighwavenumberwithcapabilitiestouseCartesianmeshesofrectanglesoverlappingthedomainboundary,andenrichmentoftheapproximationbywavefunctionspastedbythePUM.

Wecansummarizetheconclusions,whicharebasedontheoreticalunderstandingandoncomputationscomplementingwhatispossibletosaybythetheory,asfollows:

1.ThepollutioneffectfortheHelmholtzequationcannotbeavoided.Itisincreasingwithkanddecreasingwithincreasingpandq.Wecan,however,detectthepollutionbyusinganaposteriorierrorestimation.

2.Wehaveexponentialrateofconvergencewithrespecttowavedirections.Thisispredictedbythetheoryandholdsinalargerangeofpracticalaccuracy,asitcanbeseenfromournumericalexperiments.

3.Thenumericalintegrationsinfluencetheaccuracyofthesolution.Integrationerrorcanbedetectedbyaposterioriestimation.

4730T.Strouboulisetal./Comput.MethodsAppl.Mech.Engrg.195(2006)4711–4731

4.Roundofferrormustbeanalyzedtogetherwiththenumericalintegrationerrorbecausebotherrorshavesimilarchar-acter.Toanalyzetheinfluenceofroundofferroronlybytheconditionnumberisnotcorrect.Wewouldalsoliketomentionsomeoftheopenproblems:

1.Characterizationofthepollutioneffect.Thetheorythatwehaveisvalidonlyforsufficientlysmallh.

2.Theoreticalanalysisoftheinfluenceoftheintegrationerrorisnotavailable,aswellasdeterminationoflargesttoleranceintheadaptiveintegrationleadingtothegoaloftheanalysis.

3.Theroundoffanalyses,alongwiththeoryandpracticeoftheconstructionoftheshapefunctionwhichleadtothegoodconditioningnumber,andpossiblyforpreconditioningandeffectiveiterativesolverareopen.

Acknowledgements

ThisworkwassupportedbytheOfficeofNavalResearchunderGrantsN00014-99-1-0726(PI:T.Strouboulis)and

ˇka).ThesupportofDr.LuiseCouchmanoftheOfficeofNavalResearchisgreatlyN00014-99-1-0724(PI:I.Babus

appreciated.

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